Convergence exponential

Figure 4-7 The SCF energy of the neon atom converges exponentially with the number of Gaussian primitive basis functions.

Proposition 4. Since the AD model (2) is a minimum-phase system and has a well-defined relative degree r under NOC. Then, the following input-output linearizing controller will make converge exponentially the total concentration of organic substrate St to a desired value for all t > 0... 

Realizing that Eq. (13) gives an explicit solution of (1) with an appropriate V, in terms of logarithmic derivatives, it is possible to identify u with the well-known Jost solution denoted as/(r, 2), see more below and Ref. [44], which here must be proportional to the Weyl s solution x(f, )- With this identification, we obtain the generalized Titchmarsh formula (generalized since it applies to all asymptotically convergent exponential-type solutions commensurate with Weyl s limit point classification)... 

Although the expansion in Eq. (43) converges exponentially once / > a AH, it is not necessary to converge it for the calculation of eigenstates. Instead, we use the truncated polynomials as the filter in the Lanezos iteration, i.e. 

In order to introduce a compactification of a three-dimensional, quantum chemical electron density that, in principle, is neither finite nor bounded, we first shall consider a two-dimensional example, illustrated in Figure 1. Note that compactness is a generalization of the properties finite and bounded. Consider a plane P, supplied with a coordinate system and a sphere. of center c and of finite radius that is placed on the plane with a tangential contact with the plane at the origin of the coordinate system. Furthermore, assume that an almost everywhere continuous and infinitely differentiable (smooth) function F x) is specified over this plane that converges exponentially to zero at infinity. Whereas the plane is neither bounded... 

The Chebychev method converges exponentially with the number of expansion terms n for a given step size A and is particularly advantageous and efficient when A is large. However, unlike short-time propagators such as SOD or SP, the Chebychev method is not directly applicable to time-dependent or non-Hermitian Hamiltonians. 

The expansion (b) has not been studied in a formal way. A rather detailed unpublished investigation of R. Franke and one of the present authors on the Hj ion has shown numerically [162] that the error of an expansion in atomic s,p,d etc. functions appears to converge exponentially. There is some evidence that this holds for all one-electron problems, especially for Hartree-Fock. It is, for not too large molecules, not too difficult to saturate a basis and to get sufficiently close to the Hartree-Fock limit at least for the energy. 

The latter integral is bounded since both

[Pg.258]

From the equations above, we observe that state Xj tends to diverge to 1 whereas state X2 converges exponentially to 0. Therefore, there exists a set of states M e Ry which pass the point (<7y, at a certain time t = T. The set M is represented by the following state equations, with initial conditions Xi(0) and X2(0) ... 

This further implies that P(riA,t) obeys a Poisson distribution whose parameter converges exponentially (with rate y) to the stationary value X/y. 

Here, the second subscript denotes the steady state value. The roots of the quadratic characteristic equation (eigenvalues) of the matrix A determine the stability of the equations the system will converge exponentially to the steady state if all roots have a negative real part and, therefore, is asymptotically stable. It will show a limit cycle if the roots are imaginary with zero real parts. It is unstable if any of the roots has a positive real part. Since the perturbations will decay asymptotically if and only if all the eigenvalues of the matrix A have a negative real part, it follows that the necessary and sufficient conditions for local stability are ... 

Now, let the equilibrium state x = 0 of the reduced system (9.1.2) be stable in the sense of Lyapunov. By definition, this means that for the system (9.1.2 and 9.1.3) in the standard form, the x-coordinate remains small in the norm for all positive times, for any trajectory which starts sufficiently close to O, provided y remains small. At the same time, the smallness of x implies the inequality (9.1.4) for the y-coordinate, i.e. y t) converges exponentially to zero. Thus, we have the following theorem. 

Since the dynamics in the y-variables is trivial — they converge exponentially to the origin, it suffices for us to consider only bifurcations in the restriction of the system (11.3.2) to the center manifold ... 

The range of values of x for which each of the series is convergent is stated at the right of the series. 2.2.4.1 Exponential and Logarithmic Series... 

Quadralically Convergent or Second-Order SCF. As mentioned in Section 3.6, the variational procedure can be formulated in terms of an exponential transformation of the MOs, with the (independent) variational parameters contained in an X matrix. Note that the X variables are preferred over the MO coefficients in eq. (3.48) for optimization, since the latter are not independent (the MOs must be orthonormal). The exponential may be written as a series expansion, and the energy expanded in terms of the X variables describing the occupied-virtual mixing of the orbitals. 

The percolation theory [5, 20-23] is the most adequate for the description of an abstract model of the CPCM. As the majority of polymers are typical insulators, the probability of transfer of current carriers between two conductive points isolated from each other by an interlayer of the polymer decreases exponentially with the growth of gap lg (the tunnel effect) and is other than zero only for lg < 100 A. For this reason, the transfer of current through macroscopic (compared to the sample size) distances can be effected via the contacting-particles chains. Calculation of the probability of the formation of such chains is the subject of the percolation theory. It should be noted that the concept of contact is not just for the particles in direct contact with each other but, apparently, implies convergence of the particles to distances at which the probability of transfer of current carriers between them becomes other than zero. 

The optimum interval length goes as l/v nand the error as exp (—7T /n). This is certainly a much faster convergence than for the choice of an equidistant grid for the exponential function as studied in appendix B. 

---